Web of dualities on non-orientable surfaces

TL;DR

This work analyzes a web of dualities in two-dimensional quantum field theories with a non-anomalous $\mathbb{Z}_2$ symmetry and time-reversal, showing that the full set of topological manipulations generated by gauging and fermionization form the dihedral group $D_8$ of order 16. It develops a unified framework via Symmetry TFT to interpret these manipulations as domain walls and gapped boundary data in a bulk $\mathbb{Z}_2$ gauge theory, and then translates the dualities into the language of $S^1$-sector decompositions of the Hilbert space, clarifying how bosonic and fermionic theories transform among untwisted/twisted and parity sectors. The paper provides explicit formulas for the manipulations in both bosonic and fermionic theories, and demonstrates the structure with the Majorana/Ising CFT as a concrete example, including a lattice realization and a detailed character-level correspondence. Overall, the results illuminate how topological operations organize into $D_8$ and reveal precise sector-by-sector dualities that connect spin/Pin$^-$-dependent fermionization with conventional gauging, advancing the understanding of dualities on non-orientable surfaces and their physical consequences.

Abstract

It is known that a two-dimensional bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry can be fermionized. Recent work shows that if the bosonic theory also has non-anomalous time-reversal symmetry, fermionization extends to non-orientable surfaces and yields a fermionic theory that depends on a $\mathrm{Pin}^-$ structure. Besides fermionization, one can define various topological manipulations, such as gauging and stacking invertible phases, which together generate a web of dualities. We prove that their group structure is the dihedral group $D_8$ of order 16. Furthermore, we systematically investigate the web from two perspectives: Symmetry TFT and actions on sectors of the $S^1$ Hilbert space.

Figures (5)

• Figure 1: A web of dualities between fermionization and gauging.
• Figure 2: The generic orbit of $D_8$. In a fermionic theory, $n,n'$ denote $T_{F,n},T'_{F,n}$, respectively.
• Figure 3: Symmetry TFT provides a framework for understanding the partition functions $Z_{T_B}[a]$ as inner products between a physical (dynamical) boundary state $\ket{T_B}$ and a gapped (topological) boundary state $\bra{e,a}$, possibly with the insertion of a domain wall defect along the interval.
• Figure 4: An operation on the theory is described as attaching a domain wall from the bulk that implements the operation.
• Figure 5: The torus $T^2$ is represented as a cylinder with its ends identified, with time running upward. The Poincaré duals of the spatial and temporal cycles define $x,y \in H^1(T^2;\mathbb{Z}_2)$. For the Klein bottle, an orientation-reversing defect is inserted along the spatial direction, which affects the computation. See \ref{['int on K']}.